I have a dataset of medical centers and I need to predict their infection rate, based on the center characteristics and aggregated patient data (eg. percentage of patients which underwent a certain procedure). For each center I have the actual list of patient level data, but just for the cases (pts with infection), not for the controls.
I don't know whether it's better to use center level data, with the logit scale infection risk as outcome, weighted by the total number of patients in the center or replicate center features at the patient level using the real cases and faking the controls, in order to have a yes/no outcome.
The problem arises from the fact that smaller centers have extreme (high or low) risk levels that are evidently random fluctuations, so I cannot use the center level data without adjusting for this. The replication would solve this problem, but the dataset becomes huge and computationally intensive for ML training. If I use the weights I don't know how to split the data in train and test correctly. Furthermore, the size (n. of beds) of the center is used as predictors and I don't know how the ML algorithm would manage the increased variability in smaller hospital if there's a predictor associated to it.